Problem 16 - Elementary Algebra

Question

Find the equation of the line that contains the two given points. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective \(1 \mathrm{~b}\) ) \((-2,8)\) and \((4,-2)\)

Step-by-Step Solution

Answer

The equation of the line is \(5x + 3y = 14\).

Step 1: Find the Slope

First, calculate the slope of the line passing through the points \((-2, 8)\) and \((4, -2)\). The formula for the slope \(m\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points, compute \(m = \frac{-2 - 8}{4 - (-2)} = \frac{-10}{6} = -\frac{5}{3}\).

Step 2: Use Point-Slope Form

With the slope \(m = -\frac{5}{3}\) and one of the points, use the point-slope form of a line equation: \(y - y_1 = m(x - x_1)\). Using the point \((-2, 8)\), the equation becomes \(y - 8 = -\frac{5}{3}(x + 2)\).

Step 3: Convert to Standard Form

Expand and simplify the point-slope equation: \(y - 8 = -\frac{5}{3}x - \frac{10}{3}\). Multiply every term by 3 to eliminate the fraction: \(3y - 24 = -5x - 10\), which simplifies to \(5x + 3y = 14\).

Step 4: Ensure Integer Coefficients

The simplified equation, \(5x + 3y = 14\), is already in the standard form \(Ax + By = C\), with integer coefficients 5, 3, and 14.

Key Concepts

Slope

To find the equation of a line between two points, the first step is to determine the slope. The slope of a line measures the steepness or inclination between two points on the line. It's denoted by the letter 'm'. A positive slope means the line goes upwards as you move from left to right, and a negative slope means it goes downwards.
To calculate the slope, use the formula:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• Here, \((x_1,y_1)\) and \((x_2,y_2)\) are the coordinates of the two given points.
• In our case, the points are \((-2, 8)\) and \((4, -2)\).
• Substituting these values into the formula gives:

\[ m = \frac{-2 - 8}{4 - (-2)} = \frac{-10}{6} = -\frac{5}{3} \]This calculation shows that the slope of the line is \(-\frac{5}{3}\). It's important to simplify the fraction for accuracy and clarity.

Point-Slope Form

Once the slope has been calculated, the Point-Slope Form is a helpful way to write the equation of a line. This form particularly comes in handy when you have a point on the line and the slope. The Point-Slope Form formula is:

• \[ y - y_1 = m(x - x_1) \]
• Here, \(m\) stands for the slope, and \((x_1, y_1)\) is a point on the line.
• For this problem, using either point \((-2, 8)\) or \((4, -2)\) will work, but let's use \((-2, 8)\) and the slope \(-\frac{5}{3}\).

Substitute into the formula:\[y - 8 = -\frac{5}{3}(x + 2)\]This equation represents the line in Point-Slope Form. Working with this form allows you to clearly see the relationship between the slope and a point on the line. If necessary, it can easily be converted into other forms like the Standard Form.

Standard Form Equation

The Standard Form Equation of a line is another common way to express the equation of a line. It's written in the format of:

• \[ Ax + By = C \]
• Here, \(A\), \(B\), and \(C\) are integers, and \(x\) and \(y\) are variables representing any point on the line.

In the solution, we begin with the Point-Slope Form equation:\[ y - 8 = -\frac{5}{3}(x + 2) \]To convert this into Standard Form, first expand the right side:\[ y - 8 = -\frac{5}{3}x - \frac{10}{3} \]Multiply through by 3 to eliminate fractions:\[ 3(y - 8) = -5(x + 2) \]Simplifying:\[ 3y - 24 = -5x - 10 \]Rearranging gives us the Standard Form:\[ 5x + 3y = 14 \]With this, the coefficients are integers, fulfilling the Standard Form criteria. This representation is succinct and helpful for quickly assessing the relationship between x and y on the graph.